Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,9,16,11,17,10,39)(5,18,22,41,23,50,37,51)(6,24,28,46,29,52,38,60)(7,30,8,34,33,15,32,14)(12,21,42,20,44,35,43,31)(13,26,47,27,49,61,48,36)(19,54,57,65,58,40,25,45)(53,62,63,56,66,55,64,59);(3,5,7,31,10,22,32,21,11,23,33,20,9,37,8,35)(4,12,34,18,16,43,15,51,17,44,14,50,39,42,30,41)(6,25,61,56,28,58,36,62,29,57,26,59,38,19,27,55)(13,45,46,64,48,54,24,53,49,65,60,63,47,40,52,66);(3,6,7,26,10,28,32,27,11,29,33,61,9,38,8,36)(4,13,14,46,16,48,30,24,17,49,34,60,39,47,15,52)(5,19,20,55,22,25,35,56,23,58,31,62,37,57,21,59)(12,40,18,53,43,45,51,63,44,54,50,66,42,65,41,64)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402130);